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APPENDIX G MULTISTATE LIFE TABLE METHODOLOGY AND PROJECTIONS Multistate period life tables were used in this report to develop projections of numbers of new Ph.D.s that would be needed in the future to sustain certain growth rates of the labor force. This Appendix describes briefly the method used to generate these results.) Life table techniques are sometimes the only way to ob- tain estimates of certain statistics describing mobility and career characteristics of a population, especially those re- lated to rates of occurrence of events, duration of time spent in an activity, and rates of attrition or exit from a population (due to death, retirement, job changing, etc.) even when we have not observed the full lifetimes of the scientists with our data (which is often the case with most data sets).2 Moreover, life table methods provide a useful way of orga- nizing various age-specific rates (rates of entering the labor force, changing jobs, moving abroad, retiring, and dying) into a logical framework, which can then be used to make projections of various characteristics of a population, such as its age distribution. Multistate life tables, an extension of basic life tables, allow greater complexity to enter the analysis: people can enter as well as exit a population and can move back and forth across a variety of states within a population. Life tables may be classified as period versus cohort life tables. The Panel on Estimation Procedures decided that it was more practical, for the purposes of making projections, to use the former. Construction of period life tables involves taking age- specific transition rates (job changing, unemployment, re- tirement, and death rates) prevailing during a particular pe- riod (e.g., 1989-1991) and applying them to a hypothetical (synthetic) cohort of people (actually a "synthetic" cohort of new Ph.D.s). Probabilities are then calculated of en- trance or exit from a state (probability of entering a postdoctoral position, for example) and length of time spent 149 in various states, as implied by the life table. Statistics of interest were developed for three time periods (1985-1991, 1979-1985, 1973-1979~. Period life table results are con- ventionally referred to as "expected" quantities (i.e., "ex- pected fraction of people who...,", "expected length of time..") because of the nature of the methodology: con- structing a single hypothetical Ph.D. cohort that experiences the current transition rates taken from a variety of Ph.D. cohorts. The data for the life table analysis come from the longi- tudinal Survey of Doctorate Recipients (SDR), a sample survey that follows a group of Ph.D.s over time, interview- ing basically the same people every 2 years. A general description follows (for details, see NRC, 1991 SDR Meth- odological Report, forthcoming). In 1973 an initial sample of science or engineering Ph.D.s living in the United States was drawn, and those sample members have been followed through time. New Ph.D.s enter the SDR in 1975 and each subsequent SDRyear(1977, 1979,...1991)andarefollowed over time as well. Individuals are followed until they reach a certain cutoff point that depends on the survey year at which they entered [typically 42 years after the Ph.D., al- though in recent SDR waves, Hey are followed until they reach age 70 or until they drop out for over reasons (non- response or deathly. The form of the data on which the life tables are based consists mostly of large sets of transition tables constructed from the SDR by National Research Council (NRC) staff (but death rates are obtained from TIAA-CREF data from the late 1980s, taken from Bowen and Sosa, 19914. For every pair of biennial survey-interview years ("waves") in the SDR (1973-1975 as the first pair, 1989-1991 as the last pair), the number of people moving between various states within those 2 years were obtained. These states were: postdoctorate, R&D employments within one' s broad Ph.D.

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APPENDIX G field (biomed, behavioral, other), non-A&D employment within broad Ph.D. field, employment outside of broad Ph.D. field, out of labor force or unemployed (combined), leaving the country, retirement, and death. All of the bien- nial transition proportions were obtained by 2 year age group by broad Ph.D. field, and by sex. The survey obser- vations (i.e., people) in one set of biennial transitions are often We same people in subsequent sets Though older).4 Another data ingredient for the life tables is the distribu- tion of states (same as above) of "new entrants to the SDR" for SDR waves 1975-1991, again by age, sex, and Ph.D. field. These are used as estimates of numbers of new Ph.D.s in each survey year. These transition data sets constructed by NRC staff were transformed into proportions to be used as input into a Multistate Life Table program (Tiemeyer and Ulmer,1991~. Initial work involved explorations of data quality, sample sizes, and the stability of rates over time. To have large enough sample sizes for (what we would hope to be) reli- able estimates of sex differences in career patterns as well estimates of how Me career patterns have changed over time, it was necessary to aggregate Me data into three broad time periods (as opposed to looking at a larger number of time periods): 1985-1991, 1979-1985, 1973-1979. Projection Models Life table construction begins by calculating a matrix containing Me proportion of individuals exiting an origin state for each possible destination state between ages x and x+2 (in our case). This matrix is called Mx . Our projection models hold the population of those em- ployed "in field" to some constant growth rate. The follow- ing algorithm is used: 1. Survive the current specified Ph.D. population for- ward 2 years. 2. Calculate the number of individuals employed "in field". 3. Calculate the differences between the target "in field" population and the number "in field" in Me survived current population. This yields the number of new entrants needed to increase the "in field" population to its target size. 4. Divide the result of (3) by the proportion of new en- trants who enter an "in field" employment state on receiv- ing their Ph.D.. 5. Use Me result of (4) as Me number of new entrants who would have had to enter Me population between year y and y+2 to attain Me target "in field" population. Add these individuals into the life table, distributed approximately by age and destination state. Let Nx y represent the number of individuals in the speci- fied Ph.D. population in each employment state at age X for 150 a given year Y. Nx y is a k by k matrix, where k equals the number of states in Me model. The columns indicate origin states (in Me base year) and Me rows destination states. So Nx,l995~4,1] would equal Me number of people who were in the 4th state (out of field employment) in 1995 who were in the 1st state (in field post-doe) in 1991. For the base year, the off-diagonal elements of Nx 1991 are all 0 and the on- diagonal elements are equal Me number of individuals age X in 1991 in the specified Ph.D. population in each employ- ment state. Let N-x y represent the number of individuals in each state (by origin state in 1991) in year y, BEFORE new entrants between year y and y-2 are added into Me life table. Then N-x y is given by: N- N (I+ MX) 1 (I MX ) Let F199l represent the total number of individuals in the specified Ph.D. population employed "in field" in year 1991. Then F199l is given by: 71 3 ~ Flgg1 = ~ ~ ~ NX,l991 [d, o] x=25 o=1 d7=1 where x represents age, o represents origin of state, d repre- sents destination state, Nxl'9lLo,d] represents the dth row and the oth column of Nx 1991' and where states 1 Trough 3 represent the employed "in field" states. Let F-y represent the total number of individuals in the specified Ph.D. population employed "in field" in year Y who were in the specified Ph.D. population (although not necessarily employed "in field") in year Y-2. Then F-y is given by: 71 3 ~ By-= 77Nxy~d,O] x=25 o=1 d=1 Let G represent Me assumed 2-year growth rate for Fy. Then the target employed "in field" population size for any . . given year IS: Target "In Field" Population Size (y) = F199l (l+G)Y 1991 Let Dx represent the proportionate distribution by age and state of new entrants to the specified Ph.D. population over Me two year period between Y and Y+2. Dx is a 1 by k vector win each column representing Me proportion of all new entrants who are age X who enter that state on receiv- ing Weir Ph.D. Summing Dx across all ages and states should equal one. Finally, let R represent the proportion of all new entrants who enter an "in field" state on receiving their Ph.D. Then R is given by: 71 3 R= >~ >~DX[d] x=25 d7=1

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APPENDIX G Given NX,1991' Mx, Dx, and G. then Nx y can be calculated tional job openings Mat would need to be created to attain for any y greater than 1991 (in increments of 2-years) by the particular target rate of growth. iterating Trough the formula: In generating the estimates of job openings, the follow ing assumptions were made: Nx,yl yl991 ~ ((F199l (1+ G) ) Y ) ((S DX)xI) which expands to: Nx yly,l99l = NX y-2 (I + 2X ~ (1 _ 2x ~ + (Flg'1 (1+G)Y )- ~ I1;(NX,Y-2 (I+ 2-~ (I- 2~)[d,o] R ((S DX)xI) where I is a k by k identity matrix, S is a k by 1 vector of ones, and the symbol x designates an element-wise matrix multiplication operation. (The operation ((S D) x I) merely takes Me Dx vector and turns it into a matrix with Me ele- ments of Dx on the main diagonal and zeros on the off di- agonals). The total number of new entrants is given by the compo- nent: ( x=25 a=! d=~( ( 2 ) ( 2 )) R ((S Dx)xI) Projections were made separately for each of 4 populations: biomedical Ph.D.'s in biomedical employment fields, non-biomedical Ph.D.'s in biomedical employment fields, behavioral Ph.D.'s in behavioral employment fields, and non-behavioral Ph.D.'s in behavioral employment fields. To illustrate the use of life table analysis in generating projections of workforce variables, Me Panel, as an explor- atory exercise, chose to generate estimates of job openings. Given We uncertainty associated with efforts to project de- mand, Me Panel examined three growth rate scenarios based on the average annual growth in the biomedical and behav- ioral science workforces between 1981 and 1991: zero growth; one-half the 1981-1991 average annual growth; and the average annual growth.5 Estimates of "net separations"6 were generated using the life tables. Estimates of needed job openings for Me alternative growth scenarios were de- rived by adding to these separations the number of addi 151 1. There is never a negative number of new entrants. If there is a surplus in the employed "in field" population at a given year, no new entrants are added to the life table for that year. 2. The ratio of behavioral Ph.D.'s to non-behavioral Ph.D.'s employed in behavioral fields remains constant. That is, both Ph.D. populations increase or decrease at the same rate. The same assumption is made for Me models of biomedical and non-biomedical Ph.D.'s in biomedical em- ployment. 3. The age/destination state proportionate distribution, Dx, is taken from the age/destination distribution observed among new entrants between 1985 and 1991. 4. The age/origin state distribution for the current popu- lation, Nx An, is calculated by taking the age-specific ori- gin state distribution among Me current Ph.D. population between 1985 and 1991 and applying it to the age dis~ibu- tion of the 1991 current population. 5. The age-specific 2-year transition proportions, Mx, used to survive the current age-dis~ibution is taken from Me observed transition proportions between 1985 and 1991. The comm~ttee's Panel on Estimation Procedures will extend this work and will prepare a separate report for re- lease in 1994 on the role of multistate life table methods in the estimation of national need. NOTES 1. Methodological detail is available on request from NRC/OSEP Studies and Surveys Unit (Memorandum by Peter Tiemeyer, September 30, 1993). A general discussion of multistate life tables can be found in Keyfitz (1985). 2. In our particular project, however, we began with transition rates as the basic input data, and derived other life table statistics from those rates. 3. We define R&D employment to be basic or applied research, man- agement of R&D, or development and design of systems and products; it is based on the individual's self-report of primary or secondary work activ- ity. 4. With respect to the treatment of missing data: in general, to enter into the calculation of a biennial transition table, an individual case was required to have valid survey data on age and Ph.D. field and valid data for both of the survey years (for that transition table) on employment field (biomedical, behavioral, etc.) and employment status (postdoctorate, em- ployed, retired, etc.). We developed decision rules for the treatment of all of these variables to handle various conditions (available on request). For example, work activity (i.e., R&D vs. non-A&D) could be missing if the person's employment field was other than biomedical or behavioral (be- cause one of the "states" of the model is "employed outside of Ph.D. field" and those who are out of labor force, retired, or out of the country could be missing employment field and "work activity".

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APPENDIX G 5. The 1981-1991 average annual growth rates were: 4.25 percent per year for the biomedical sciences workforce and 3.5 percent per year for the behavioral sciences workforce. 6. Net separations are defined in this analysis as losses arising from death, retirement or outmobility to another state minus gains from inmobility of experienced scientists from other states of employment. Al- ternative definitions will be explored in subsequent work by the Panel on Estimation Procedures. REFERENCES Bowen, W.G. and J.A. Sosa 1989 Prospectsfor Faculty in theArts and Sciences. Princeton, NJ: Princeton University Press. 152 Keyfitz, N. 1985 Applied Mathematical Demography. 2nd Ed. New York: Springer-Verlag. National Research Council Forthcoming 1991 SDR Methodological Report. Washington, D.C.: National Academy Press. Tiemeyer, P. and G. Ulmer 1991 MSLT: A Program for the Computation of Multistate Life Tables. Center for Demography Working Paper 91-34, Uni- versity of Wisconsin.